Secondly, since log-concave functions are co-area regular, by [1]}, the symmetric decreasing rearrangements of log-concave functions can be approximated in the strong \(W^{1,p}\) -topology by a sequence of Steiner symmetrizations. Therefore, the Orlicz-Sobolev affine ball operator \(B_{\phi }:W_0^{1,\Phi }(G)\rightarrow \mathcal {K}_o^n\)  is continuous for log-concave functions. But for the general Sobolev functions, the Orlicz-Sobolev affine ball operator may be discontinuous for the co-area irregular functions. In this paper, making use of the weak convergence of Steiner symmetrizations of \(f\in W^{1,\phi }(\mathbb {R}^n)\) , we prove that the Orlicz-Sobolev affine ball operator is weakly continuous in such a way that
there exist a convex body \(K_0\)  and a sequence of successive Steiner symmetrizations \(\lbrace f_k\rbrace _{k\ge 0}\)  of \(f\)  such that \(B_{\phi }(f_i)\)  converges to \(K_0\)  and \(K_0\subset B_{\phi }(f^{\star })\)  (see Lemma REF ).
